2014_4_StarTemperature.ppt

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郦魄历
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1 Applications of Maxwell-Boltzmann Distribution (2) 1

2 Classical Gas: Maxwell-Boltzmann Distribution f (v x,v y,v z = n ( 2π ) 3 σ exp[ v 2 x + v 2 2 y + v z ] 3 2σ 2 n mv2 exp[ 3 ( 2π k B T / m) 2k B T ] n ( 2π k B T / m) 3 exp( E k k B T ) 2

3 Classical Gas: Maxwell-Boltzmann Distribution Classical Gas: Maxwell-Boltzmann Distribution n ( 2πk B T / m) 3 exp( E k k B T f e E k /k B T Bosons: Bose-Einstein Distribution f e E k /k B T 1 Fermions: Fermi-Dirac Distribution f e (E k µ)/k B T + 1 where µ denotes certain kind of potential energy 3

4 玻色子的spin為整數可分為基本玻色子與組合玻色子(如:原子 Applications of Maxwellian-Boltzmann 其中基本玻色子(Elementary bosons)為傳遞作用力的粒子如光子(photon)傳遞電磁力 Distribution (2) 推廣到光與低溫物理膠子(gluon)傳遞強作用力 Classical Gas: Maxwell-Boltzmann Distribution W,Z玻色子傳遞弱作用力 n Ek f 重子(graviton)傳遞重力尚未發現 f (Ek exp( Ek / k B T 3 也就是 Higgs k BT e ( 2π kb(我不懂) T / m ) 尚未發現最近被發現了今年213年的諾貝爾獎玻色子同性相吸 gays! Bosons: Bose-Einstein Distribution f (Ek f e Ek / k B T 1 Fermions: Fermi-Dirac Distribution f (Ek f e(ek µ )/ kb T + 1 where µ denotes certain kind of potential energy 4

5 費米子的spin為半整數也可分為基本費米子(elementary Applications of Maxwellian-Boltzmann fermions)如 Distribution (2) 推廣到光與低溫物理電子(electron) 微中子(neutrinos) Classical Gas: Maxwell-Boltzmann Distribution 夸克子(quarks) n Ek f f (Ek exp( Ek / k B T 與組合費米子(Composite 3 fermions)如 k BT e 質子(proton)( 2π kbt / m ) 中子(neutron) 費米子是組成物質的元素 Bosons: Bose-Einstein Distribution 費米子同性相斥(guys!) 所以溫度越低密度應該越少除非有 f f (Ek Ek / kb T 很大的束縛力(如重力) 才能製造低溫高密度的費米子簡併態 e 1 Fermions: Fermi-Dirac Distribution f (Ek f e(ek µ )/ kb T + 1 where µ denotes certain kind of potential energy 5

6 Classical Gas: Maxwell-Boltzmann Distribution Classical Gas: Maxwell-Boltzmann Distribution f e (E k µ)/k B T Bosons: Bose-Einstein Distribution f e (E k µ)/k B T 1 Fermions: Fermi-Dirac Distribution f e (E k µ)/k B T + 1 where µ denotes certain kind of potential energy 6

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8 Applications of Maxwell-Boltzmann Distribution Classical Gas: Maxwell-Boltzmann Distribution f e E k /k B T Bosons: Bose-Einstein Distribution f (ε f e ε /k BT 1 where ε = hν = h c λ = h cn 2L and n = n 2 x + n 2 2 y + n z Summing over Modes (for both left-hand and right-hand polarized waves). Let f = 1. It yields U = 2 ny n x ε f (ε n z n x ny n z hcn L 1 e hcn/2 Lk BT 1 8

9 Applications of Maxwell-Boltzmann Distribution Summing over Modes U = dn x = dn π /2 dn y = dn 4πn2 8 ndθ For ε = h cn it yields dε = hc and 2L 2L dn hcn L dn z ( hcn 1 L e hcn/2 LkBT 1 ) π /2 hcn 1 nsinθ dφ L e hcn/2 LkBT 1 1 e hcn/2 LkBT 1 U = L 3 dε 8πε 3 / (hc) 3 e ε/kbt 1 9

10 Applications of Maxwell-Boltzmann Distribution Summing over modes U = L 3 dε 8πε 3 / (hc) 3 e ε/kbt 1 Let the energy density to be u(ε), such that total energy per unit volume ( V = L 3 ) becomes U V = U L = dε 8πε 3 / (hc) 3 = u(ε)dε 3 e ε/kbt 1 Thus, we have obtain the Planck Spectrum (the energy density per unit photon energy), i.e., u(ε 8πε 3 / (hc) 3 e ε /k BT 1 1

11 Let Applications of Maxwell-Boltzmann Distribution x = ε / k B T The total energy per unit volume becomes U V = dε 8πε 3 / (hc) 3 = k e ε/kbt B T dx 8π x3 (k B T / hc) 3 1 e x 1 = 8π (k BT ) 4 dx x3 (hc) 3 e x 1 Since It yields dx x3 = π 4 e x 1 15 U V = 8π 5 (k B T ) 4 15(hc) 3 Blackbody Radiation: power per unit area c 4 ( ) The Stefan-Boltzmann constant σ = 2π 5 4 k B W = h c U V = 2π 5 (k B T ) 4 15h 3 c 2 = σt 4 m 2 K 4 (see pp.31-32) 11

12 Planck Spectrum (e.g., Figures 7.19 & 7.2 on pages ) The energy density per unit photon energy u(ε 8πε 3 / (hc) 3 e ε /k BT 1 Home Work 3.1: Plot the Planck spectrum at different temperature (e.g., T=1ºK, 5ºK, 1ºK) u(ε) The the energy density of the light per unit frequency u(ν 8πν 3 / c 3 e hν /k BT 1 h Home Work 3.2: Plot the Planck spectrum at different temperature (e.g., T=1ºK, 5ºK, 1ºK) u(ν) The the energy density of the light per unit wavelength u(λ 8π / λ 3 e hc/λk BT 1 hc λ 2 Home Work 3.3: Plot the Planck spectrum at different temperature (e.g., T=1ºK, 5ºK, 1ºK) u(λ) 12

13 Applications of Blackbody Radiation & Planck Spectrum Scientists use the blackbody radiation or the Planck spectrum to determine the temperature of a distant star. The blackbody radiation model is not applicable to a diluted medium For a diluted medium, the spectra usually show discrete emission lines from the medium. (example: aurora) 13

14 Outlines of Thermal Dynamics 焓熵 14